# Properties

 Label 441.2.a.f Level $441$ Weight $2$ Character orbit 441.a Self dual yes Analytic conductor $3.521$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(1,441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$3.52140272914$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - 2 q^{5} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 - 2 * q^5 - 3 * q^8 $$q + q^{2} - q^{4} - 2 q^{5} - 3 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{13} - q^{16} - 6 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - q^{25} + 2 q^{26} + 2 q^{29} + 5 q^{32} - 6 q^{34} + 6 q^{37} - 4 q^{38} + 6 q^{40} + 2 q^{41} - 4 q^{43} + 4 q^{44} - q^{50} - 2 q^{52} - 6 q^{53} + 8 q^{55} + 2 q^{58} + 12 q^{59} + 2 q^{61} + 7 q^{64} - 4 q^{65} + 4 q^{67} + 6 q^{68} + 6 q^{73} + 6 q^{74} + 4 q^{76} - 16 q^{79} + 2 q^{80} + 2 q^{82} - 12 q^{83} + 12 q^{85} - 4 q^{86} + 12 q^{88} - 14 q^{89} + 8 q^{95} - 18 q^{97}+O(q^{100})$$ q + q^2 - q^4 - 2 * q^5 - 3 * q^8 - 2 * q^10 - 4 * q^11 + 2 * q^13 - q^16 - 6 * q^17 - 4 * q^19 + 2 * q^20 - 4 * q^22 - q^25 + 2 * q^26 + 2 * q^29 + 5 * q^32 - 6 * q^34 + 6 * q^37 - 4 * q^38 + 6 * q^40 + 2 * q^41 - 4 * q^43 + 4 * q^44 - q^50 - 2 * q^52 - 6 * q^53 + 8 * q^55 + 2 * q^58 + 12 * q^59 + 2 * q^61 + 7 * q^64 - 4 * q^65 + 4 * q^67 + 6 * q^68 + 6 * q^73 + 6 * q^74 + 4 * q^76 - 16 * q^79 + 2 * q^80 + 2 * q^82 - 12 * q^83 + 12 * q^85 - 4 * q^86 + 12 * q^88 - 14 * q^89 + 8 * q^95 - 18 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 −1.00000 −2.00000 0 0 −3.00000 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.f 1
3.b odd 2 1 147.2.a.a 1
4.b odd 2 1 7056.2.a.p 1
7.b odd 2 1 63.2.a.a 1
7.c even 3 2 441.2.e.b 2
7.d odd 6 2 441.2.e.a 2
12.b even 2 1 2352.2.a.v 1
15.d odd 2 1 3675.2.a.n 1
21.c even 2 1 21.2.a.a 1
21.g even 6 2 147.2.e.b 2
21.h odd 6 2 147.2.e.c 2
24.f even 2 1 9408.2.a.m 1
24.h odd 2 1 9408.2.a.bv 1
28.d even 2 1 1008.2.a.l 1
35.c odd 2 1 1575.2.a.c 1
35.f even 4 2 1575.2.d.a 2
56.e even 2 1 4032.2.a.k 1
56.h odd 2 1 4032.2.a.h 1
63.l odd 6 2 567.2.f.b 2
63.o even 6 2 567.2.f.g 2
77.b even 2 1 7623.2.a.g 1
84.h odd 2 1 336.2.a.a 1
84.j odd 6 2 2352.2.q.x 2
84.n even 6 2 2352.2.q.e 2
105.g even 2 1 525.2.a.d 1
105.k odd 4 2 525.2.d.a 2
168.e odd 2 1 1344.2.a.s 1
168.i even 2 1 1344.2.a.g 1
231.h odd 2 1 2541.2.a.j 1
273.g even 2 1 3549.2.a.c 1
336.v odd 4 2 5376.2.c.l 2
336.y even 4 2 5376.2.c.r 2
357.c even 2 1 6069.2.a.b 1
399.h odd 2 1 7581.2.a.d 1
420.o odd 2 1 8400.2.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 21.c even 2 1
63.2.a.a 1 7.b odd 2 1
147.2.a.a 1 3.b odd 2 1
147.2.e.b 2 21.g even 6 2
147.2.e.c 2 21.h odd 6 2
336.2.a.a 1 84.h odd 2 1
441.2.a.f 1 1.a even 1 1 trivial
441.2.e.a 2 7.d odd 6 2
441.2.e.b 2 7.c even 3 2
525.2.a.d 1 105.g even 2 1
525.2.d.a 2 105.k odd 4 2
567.2.f.b 2 63.l odd 6 2
567.2.f.g 2 63.o even 6 2
1008.2.a.l 1 28.d even 2 1
1344.2.a.g 1 168.i even 2 1
1344.2.a.s 1 168.e odd 2 1
1575.2.a.c 1 35.c odd 2 1
1575.2.d.a 2 35.f even 4 2
2352.2.a.v 1 12.b even 2 1
2352.2.q.e 2 84.n even 6 2
2352.2.q.x 2 84.j odd 6 2
2541.2.a.j 1 231.h odd 2 1
3549.2.a.c 1 273.g even 2 1
3675.2.a.n 1 15.d odd 2 1
4032.2.a.h 1 56.h odd 2 1
4032.2.a.k 1 56.e even 2 1
5376.2.c.l 2 336.v odd 4 2
5376.2.c.r 2 336.y even 4 2
6069.2.a.b 1 357.c even 2 1
7056.2.a.p 1 4.b odd 2 1
7581.2.a.d 1 399.h odd 2 1
7623.2.a.g 1 77.b even 2 1
8400.2.a.bn 1 420.o odd 2 1
9408.2.a.m 1 24.f even 2 1
9408.2.a.bv 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{5} + 2$$ T5 + 2 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T - 2$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T - 2$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T + 16$$
$83$ $$T + 12$$
$89$ $$T + 14$$
$97$ $$T + 18$$